The present invention relates generally to systems and methods for mapping, and specifically to methods of mapping of intrabody organs.
Cardiac mapping is used to locate aberrant electrical pathways and currents within the heart, as well as mechanical and other aspects of cardiac activity. Various methods and devices have been described for mapping the heart. Such methods and device are described, for example, in U.S. Pat. Nos. 5,471,982 and 5,391,199 and in PCT patent publications WO94/06349, WO96/05768 and WO97/24981. U.S. Pat. No. 5,391,199, for example, describes a catheter including both electrodes for sensing cardiac electrical activity and miniature coils for determining the position of the catheter relative to an externally-applied magnetic field. Using this catheter a cardiologist may collect a set of sampled points within a short period of time, by determining the electrical activity at a plurality of locations and determining the spatial coordinates of the locations.
In order to allow the surgeon to appreciate the determined data, a map, preferably a three dimensional (3D) map, including the sampled points is produced. U.S. Pat. No. 5,391,199 suggests superimposing the map on an image of the heart. The positions of the locations are determined with respect to a frame of reference of the image. However, it is not always desirable to acquire an image, nor is it generally possible to acquire an image in which the positions of the locations can be found with sufficient accuracy.
Various methods are known in the art for reconstructing a 3D map of a cavity or volume using the known position coordinates of a plurality of locations on the surface of the cavity or volume. Some methods include triangulation, in which the map is formed of a plurality of triangles which connect the sampled points. In some cases a convex hull or an alpha-hull of the points is constructed to form the mesh, and thereafter the constructed mesh is shrunk down to fit on the sampled points within the hull. Triangulation methods do not provide a smooth surface and therefore require additional stages of smoothing.
Another method which has been suggested is forming a bounding ellipsoid which encloses the sampled points. The sampled points are projected onto the ellipsoid, and the projected points are connected by a triangulation method. The triangles are thereafter moved with the sampled points back to their original locations, forming a crude piecewise linear approximation of the sampled surface. However, this method may reconstruct only surfaces which have a star shape, i.e., a straight line connecting a center of the reconstructed mesh to any point on the surface does not intersect the surface. In most cases heart chambers do not have a star shape.
In addition, reconstruction methods known in the art require a relatively large number of sampled locations to achieve a suitable reconstructed map. These methods were developed, for example, to work with CT and MRI imaging systems which provide large numbers of points, and therefore generally work properly only on large numbers of points. In contrast, determining the data at the locations using an invasive catheter is a time-consuming process which should be kept as short as possible, especially when dealing with a human heart. Therefore, reconstruction methods which require a large number of determined locations are not suitable.
It is an object of the present invention to provide an improved method for creating a map of a 3D volume or cavity, based on the positions of points on a surface of the volume or cavity.
It is an object of some aspects of the present invention to provide methods and apparatus for generating a map of a volume in the human body from a plurality of sampled points, regardless of the shape of the volume.
It is another object of some aspects of the present invention to provide a simple, rapid method for reconstructing a 3D map of a volume in the human body from a plurality of sampled points, preferably using fewer sampled points than is feasible using methods known the art.
It is another object of preferred embodiments of the present invention to provide a method for reconstructing a 3D map of a volume in the human body from a plurality of sampled points, without assuming any topological relationship between the points.
It is another object of some aspects of the present invention to provide a simple method for reconstructing a 3D map of a volume in movement.
It is another object of some aspects of the present invention to provide a simple method for reconstructing a 3D map of a volume in the human body from a plurality of sampled points independent of the sampling order.
It is another object of some aspects of the present invention to provide a quick method for reconstructing a 3D map of a volume in the human body from a plurality of sampled points, such that the method may be used in interactive procedures.
It is another object of some aspects of the present invention to provide a method for reconstructing a smooth 3D map of a volume in the human body from a plurality of sampled points.
In preferred embodiments of the present invention, a processor reconstructs a 3D map of a volume or cavity in a patient""s body (hereinafter referred to as the volume), from a plurality of sampled points on the volume whose position coordinates have been determined. In contrast to prior art reconstruction methods in which a large number of sampled points are used, the preferred embodiments of the present invention are directed to reconstruction of a surface based on a limited number of sampled points. The number of sampled points is generally less than 200 points and may be less than 50 points. Preferably, ten to twenty sampled points are sufficient in order to perform a preliminary reconstruction of the surface to a satisfactory quality.
An initial, generally arbitrary, closed 3D curved surface (also referred to herein for brevity as a curve) is defined in a reconstruction space in the volume of the sampled points. The closed curve is roughly adjusted to a shape which resembles a reconstruction of the sampled points. Thereafter, a flexible matching stage is preferably repeatedly performed once or more to bring the closed curve to accurately resemble the shape of the actual volume being reconstructed. Preferably, the 3D surface is rendered to a video display or other screen for viewing by a physician or other user of the map.
In preferred embodiments of the present invention, the initial closed curved surface encompasses substantially all the sampled points or is interior to substantially all the sampled points. However, it is noted that any curve in the vicinity of the sampled points is suitable. Preferably, the closed 3D curved surface comprises an ellipsoid, or any other simple closed curve. Alternatively, a non-closed curve may be used, for example, when it is desired to reconstruct a single wall rather than the entire volume.
A grid of a desired density is defined on the curve, and adjustment of the curve is performed by adjusting the grid points. The grid preferably divides the curved surface into quadrilaterals or any other polygons such that the grid evenly defines points on the curve. Preferably, the grid density is sufficient such that there are generally more grid points than sampled points in any arbitrary vicinity. Further preferably, the grid density is adjustable according to a desired compromise between reconstruction accuracy and speed.
In some preferred embodiments of the present invention, external information is used to choose an initial closed curve which is more closely related to the reconstructed volume, for example, using the image of the volume, as described above. Thus, the reconstruction procedure may produce a more accurate reconstruction in less time. Alternatively or additionally, a database of closed curves suitable for various volumes of the body is stored in a memory, and the curve to be used is chosen according to the specific procedure. In a further preferred embodiment of the present invention, a map of a reconstructed volume in a patient is used as a beginning curve for subsequent mapping procedures performed at later times on the same volume.
Preferably, the rough adjustment of the closed curve is performed in a single iteration, most preferably by calculating for each grid point an adjustment point, and moving the grid point a fraction of the distance to the adjustment point. Preferably, the grid point is moved about 50-80% of the distance between its original point and the adjustment point, more preferably about 75%.
The adjustment point is preferably determined by taking a weighted sum over substantially all the sampled points. Preferably, the weights are inversely related to the distances from the adjusted grid point to the sampled points, referred to herein as grid distances. In a preferred embodiment of the present invention, each weight is defined as the reciprocal of the sum of a small constant plus the grid distance, raised to a predetermined power, so that sampled points close to the grid point are given a larger weight. Preferably, the power is approximately between 4 to 9, most preferably 8. The small constant is preferably smaller than the magnitude of the smallest grid distance, and is preferably of the size of the accuracy of the determination of the coordinates of the sampled points. The small constant is used to prevent division by zero when a grid-point is on a sampled point.
In some preferred embodiments of the present invention, the weights also include a factor which is indicative of the density of points in the vicinity of their corresponding point. Preferably, the weight is multiplied by a density value between zero and one, indicative of the density, such that isolated sampled points influence the sum more than sampled points in a dense area. Preferably, the influence of the points is thus substantially independent of the density of points in their vicinity.
In a preferred embodiment of the present invention, the flexible matching step is performed by associating each sampled point with a corresponding grid-point, such that each sampled point is associated with the grid point which is closest to it. A movement vector is calculated for each of the associated and non-associated grid-points. Preferably, the movement vectors are calculated based on vectors from the associated grid points to their respective sampled points. Further preferably, the sampled points influence the value of the movement vector for a specific point according to their proximity to the specific point. In addition, the function by which the movement vectors are calculated is preferably smooth and does not include complicated calculations. Preferably, the function is a weighted sum of the vectors from the associated grid points to their respective sampled points. The grid points are then moved according to their respective movement vectors.
Additionally or alternatively, the associated grid points are moved toward their corresponding sampled points by a percentage of the distance between them. Those grid points which are not associated with a sampled point are moved a distance which is determined by interpolation between the distances which surrounding points on the grid are moved. Preferably, the resulting grid is smoothed using a suitable smoothing transformation. Preferably, the process of associating and moving is repeated two or more times to allow finer adjustment of the closed curve.
In a preferred embodiment of the present invention, a user can adjust the number of times the flexible matching step is repeated according to a desired compromise between image quality and speed. Alternatively or additionally, a quick reconstruction is first provided to the user, and thereafter the calculation is repeated to receive a finer reconstruction. Preferably, the weights of the weighted sum used in the flexible matching stage are adjusted according to the number of times the matching is to be performed. Alternatively or additionally, the weights are determined for each flexible matching step according to its place in the sequential order of the flexible matching steps.
Preferably, the distances used for the weights and/or for interpolation are Euclidean geometrical distances between the points. The Euclidean distance is easily computed and causes points on opposite walls of the volume to mutually repel, so that the walls do not intersect. Alternatively, other distances, such as the distance along the original or adjusted grid, may be used. In a preferred embodiment of the present invention, during the first flexible matching step the distance used is the distance along the original grid while subsequent flexible matching steps use the Euclidean distance.
In some preferred embodiments of the present invention, a smoothing process is applied to the reconstructed surface, preferably by applying a surface convolution with a Gaussian-like kernel. The smoothing process provides a better approximation of the surface and allows easier performance of calculations based on the reconstructed surface. However, applying the surface convolution results in some shrinkage of the surface, and therefore an affine transformation is preferably performed on the smoothed surface. The affine transformation is preferably chosen according to those sampled points which are external to the reconstructed surface. The chosen affine transformation preferably minimizes the mean square distance of the external points to the surface.
Preferably, when the reconstruction is finished, each sampled point substantially coincides with a grid point. In some preferred embodiments of the present invention, a final exact matching stage is performed. Each sampled point is associated with a closest grid point, and the associated grid point is moved onto the sampled point. The rest of the grid points are preferably not moved. Generally, most of the sampled points are by this stage very close to the reconstructed surface, and therefore the smoothness of the surface is substantially not affected. However, some outlier sampled points, i.e., sampled points which do not belong to the surface, may cause substantial changes to the surface. Preferably, the user may determine whether to move the surface onto points that are distanced from the surface by more than a predetermined maximum distance. Alternatively or additionally, the entire exact matching step is optional and is applied only according to a user request.
Further alternatively or additionally, the grid points are brought to a fixed distance from the sampled points. Leaving such a fixed distance may be desired, for example, when the sampled coordinates are of locations close to a distal tip of a sampling catheter rather than at the distal tip itself.
In preferred embodiments of the present invention, data regarding the sampled points are acquired by positioning a catheter within the volume which is to be reconstructed, for example, within a chamber of the heart. The catheter is positioned with a distal end thereof in contact with each of the sampled points in turn, and the coordinates of the points and, optionally, values of one or more physiological parameters are sensed at a distal end of the catheter. Preferably, the catheter comprises a coordinate sensor close to its distal end, which outputs signals indicative of the coordinates of the tip of the catheter. Preferably, the coordinate sensor determines the position by transmitting and receiving electromagnetic waves, as described, for example, in PCT publications GB93/01736, WO94/04938, WO97/24983 and WO96/05768, or in U.S. Pat. No. 5,391,199, commonly owned by the present assignee and which are all incorporated herein by reference.
In some preferred embodiments of the present invention, the reconstructed volume is in movement, for example, due to beating of the heart. In such embodiments, the sampled points are preferably registered with a reference frame fixed to the heart. Preferably, a reference catheter is fixed in the heart, and the sampled points are determined together with the position of the reference catheter which is used to register the points, as described, for example, in the above-mentioned U.S. Pat. No. 5,391,199 and PCT publication WO96/05768.
Alternatively or additionally, when at least part of the movement is a cyclic movement, as in the heart, acquisition of the sampled points is synchronized to a specific time point of the cycle. Preferably, when the sampled volume is in the heart, an ECG signal is received and is used to synchronize the acquisition of the sampled points. For example, the sampled points may be acquired at end diastole. Further alternatively or additionally, the coordinates of each of the sampled points are determined together with an indication of the time point relative to the cyclic movement in which the coordinates were acquired. Preferably, the indication includes the relative time from the beginning of the cycle and the frequency of the cyclic movement. According to the frequency and the relative time, the determined coordinates are corrected to end diastole, or any other point in the cyclic movement.
In some preferred embodiments of the present invention, for each sampled point a plurality of coordinates are determined at different time points of the cyclic movement. In one of these preferred embodiments, each sampled point has two coordinates which define the range of movement of the point. Preferably, if the plurality of coordinates of different points are associated with different cycle frequencies, the coordinates are transformed so as to correspond to a set of coordinates in a single-frequency cyclic movement. Further preferably, the coordinates are processed so as to reduce or substantially eliminate any contribution due to movement other than the specific (cardiac) cyclic movement, such as movement of the chest due to respiration. Reconstruction is performed for a plurality of configurations of the volume at different time points of the cyclic movement. Preferably, a first reconstruction is performed as described above to form an anchor reconstruction surface, and reconstruction of surfaces for other time points of the cycle are performed relative to the anchor reconstruction surface.
Preferably, for each further time point of the cyclic movement, the anchor surface is adjusted according to the coordinates of the sampled points at the further time point relative to the coordinates of the sampled points of the anchor surface. Preferably, the anchor surface is adjusted by a quadratic transformation which minimizes a mean square error, the error representing the distances between the sampled points of the further time point and the adjusted surface. Alternatively or additionally, an affine transformation is used instead of the quadratic transformation. Further alternatively or additionally, a simple transformation is used for surfaces having relatively few sampled points, while surfaces with a relatively large number of sampled points a quadratic transformation is used. The simple transformation may be an affine transformation, a scaling and rotation transformation, a rotation transformation, or any other suitable transformation.
Preferably, the adjustment of the surface for the further time points includes, after the transformation, one or more, preferably two, flexible matching steps and/or an exact matching stage.
Alternatively or additionally, the reconstruction is performed separately for each of the further time points. Further alternatively or additionally, a first reconstruction of the surfaces for the further time points is performed relative to the anchor surface, and afterwards a more accurate reconstructed is performed for each time point independently.
In some preferred embodiments of the present invention, dedicated graphics hardware which is designed to manipulate polygons is used to perform the reconstruction stages described above.
In some preferred embodiments of the present invention, one or more physiological parameters are acquired at each sampled point. The physiological parameters for the heart may comprise a measure of cardiac electrical activity, for example, and/or may comprise any other type of local information relating to the heart, as described in the PCT patent publication WO97/24981, also owned by the present assignee and further incorporated herein by reference. The one or more physiological parameters may be either scalars or vectors and may comprise, for example, a voltage, temperature, pressure, or any other desired value.
Preferably, after the volume is reconstructed based on the coordinates, values of the physiological parameter are determined for each of the grid points based on interpolation of the parameter value at surrounding sampled points. Preferably, the interpolation of the physiological parameter is performed in a manner proportional to the aggregate interpolation of the coordinates. Alternatively, the physiological parameters are interpolated according to the geometrical distance between the points on the grid. Alternatively or additionally, the physiological parameters are interpolated in a manner similar to the flexible matching step described hereinabove.
The reconstructed surface may be displayed in movement, and/or a physician may request a display of a specific time point of the cycle. Preferably, the physiological parameter is displayed on the reconstructed surface based on a predefined color scale. In a preferred embodiment of the present invention, the reliability of reconstruction of regions of the reconstructed surface is indicated on the displayed surface. Preferably, regions which are beneath a user-defined threshold are displayed as semi-transparent, using known methods such as xcex1-blending. Preferably, the reliability at any grid point is determined according to its proximity to sampled points. Those points on the grid which are beyond a predetermined distance from the nearest sampled point are less reliable.
In some preferred embodiments of the present invention, acquired images such as LV-grams and fluoroscopic images are used together with the sampled points to enhance the speed and/or accuracy of the reconstruction. Preferably, the processor performs an object recognition procedure on the image to determine the shape of the closed 3D curved surface to use in constructing the initial grid of the reconstruction. Alternatively or additionally, the image is used by the physician to select areas in which it is most desired to receive sampled points.
In some preferred embodiments of the present invention, the physician may define points, lines, or areas on the grid which must remain fixed and are not to be adjusted. Alternatively or additionally, some points may be acquired as interior points which are not to be on the map since they are not on a surface of the volume. The reconstruction procedure is performed accordingly so that the closed curve is not moved too close to the interior points.
In some preferred embodiments of the present invention, the reconstruction surface is used to determine an accurate estimate of the volume of the cavity. The surface is divided by the grid points into quadrilaterals, and each quadrilateral is further divided into two triangles. Based on these triangles the volume defined by the surface is estimated. Alternatively, the volume is calculated using a volumetric representation. Other measurements, such as geodesic surface measurements on the surface, may also be performed using the reconstructed surface.
It is noted that some of the stages described above may be ignored in some preferred embodiments of the invention, in order to save processing time and speed up the reconstruction procedure.
There is therefore provided in accordance with a preferred embodiment of the present invention, a method of reconstructing a map of a volume, including determining coordinates of a plurality of locations on a surface of the volume having a configuration, generating a grid of points defining a reconstruction surface in 3D space in proximity to the determined locations, for each of the points on the grid, defining a respective vector, dependent on a displacement between one or more of the points on the grid and one or more of the locations, and adjusting the reconstruction surface by moving substantially each of the points on the grid responsive to the respective vector, so that the reconstruction surface is deformed to resemble the configuration of the surface.
Preferably, the method includes displaying the reconstruction surface.
Preferably, generating the grid includes generating a grid such that the reconstruction surface encompasses substantially all of the determined locations or is interior to substantially all of the determined locations.
Preferably, generating the grid includes defining an ellipsoid.
Preferably, the reconstruction surface is defined and adjusted substantially independently of any assumption regarding a topology of the volume.
Further preferably, the reconstruction surface is defined and adjusted substantially without reference to any point within the volume.
Alternatively or additionally, generating the grid includes acquiring an image of the volume and defining the reconstruction surface such that it resembles the image of the volume.
Further alternatively or additionally, generating the grid includes choosing a grid from a memory library according to at least one characteristic of the volume.
Preferably, adjusting the surface includes a rough adjustment stage and a flexible matching stage.
Preferably, the rough adjustment stage includes moving each point on the grid toward a respective weighted center of mass of the determined locations, and locations closer to the point on the grid are given larger weight.
Preferably, moving each point in the rough adjustment stage includes defining, for each of the points on the grid, a respective rough adjustment vector which includes a weighted sum of vectors from the point to each of the determined locations and moving the points a distance proportional to the respective vector.
Preferably, defining the rough adjustment vector includes calculating a weight for each of the summed vectors that is generally inversely proportional to a magnitude of the summed vector raised to a predetermined power.
Preferably, the weight includes an inverse of a sum of a constant and the magnitude of the vector raised to a power between 4 and 10.
Preferably, the constant is smaller than a precision of the location determination.
Preferably, moving each point includes moving each point toward a respective target point by a distance between 50 and 90% of the distance between the point and the target point.
Preferably, the flexible matching stage includes selecting a grid point to be associated respectively with each of the determined locations. Preferably, selecting the grid point includes finding for each determined location a point on the grid that is substantially closest thereto.
Further preferably, the flexible matching stage includes moving the selected grid points toward their respective determined locations.
Preferably, moving the selected grid points includes moving the grid points substantially onto their respective, determined locations.
Preferably, the flexible matching stage includes moving grid points that were not selected by an amount dependent on the movements of surrounding grid points.
Preferably, moving the grid points that were not selected includes moving the grid points by an amount dependent substantially only on the movements of surrounding selected grid points.
Preferably, moving the grid points includes calculating a movement of a grid point that was not selected based on the movements of the surrounding selected grid points and distances from these surrounding grid points.
Preferably, calculating the movement of the grid point includes interpolating between the movements of surrounding grid points.
Preferably, the distances include geometrical distances. Alternatively or additionally, the distances include a length of the reconstruction surface between the grid points.
Preferably, the flexible matching stage includes defining a displacement function which includes a weighted sum of vectors, each vector connecting a location and its associated point.
Preferably, the flexible matching stage includes moving the grid points according to the displacement function so as to smooth the surface.
Preferably, determining the coordinates includes positioning a catheter tip at the plurality of locations.
Preferably, positioning the catheter tip includes positioning the catheter at a plurality of locations in a chamber of the heart.
Preferably, determining the coordinates includes positioning a catheter tip at the plurality of locations.
Preferably, determining the coordinates includes transmitting and receiving non-ionizing waves.
Preferably, determining the coordinates includes positioning at the plurality of locations a device which generates signals indicative of the position of the device.
Preferably, the device generates signals indicative of the six degrees of position and orientation of the device.
Preferably, determining the coordinates includes receiving the coordinates from an external source.
Preferably, the method includes acquiring a signal indicative of a value of physiological activity at substantially each of the plurality of locations.
Preferably, acquiring the signal includes acquiring a signal indicative of a value of electrical activity at the location.
Preferably, the method includes estimating a value of the physiological activity at the adjusted grid points.
Preferably, estimating the value of the physiological activity includes estimating based on an acquired value of the physiological activity at a location in a vicinity of the adjusted grid points.
Preferably, estimating based on the acquired value includes interpolating the value responsive to deformation of the reconstruction surface.
Preferably, determining coordinates of a plurality of locations includes determining coordinates of less than 200 locations, more preferably of less than 50 locations, and most preferably of less than 20 locations.
Preferably, the volume is in motion, and determining the coordinates includes determining a correction factor responsive to the motion.
Preferably, the motion includes cyclic motion, and determining the correction factor includes determining a factor responsive to a cycle frequency of the motion.
Preferably, determining the factor includes filtering out motion at a frequency substantially different from the cycle frequency.
Preferably, the motion includes cyclic motion, and determining the coordinates includes determining the coordinates at a predetermined phase of the cyclic motion.
Preferably, determining the coordinates at the predetermined phase includes determining the coordinates in a plurality of time points and adjusting the coordinates relative to the cyclic movement.
Preferably, adjusting the coordinates includes determining a rate of the cyclic movement together with the coordinates for substantially each coordinate determination.
Preferably, generating the grid and adjusting the reconstruction surface are performed separately with respect to the coordinates determined in each phase of the cyclic motion.
Alternatively or additionally, generating and adjusting are performed for the coordinates of a plurality of phases of the cyclic motion so as to form a motion map of the volume.
Preferably, generating the grid and adjusting the reconstruction surface are performed for a first group of coordinates determined in a first phase of the cyclic motion, and the reconstructed surface of the first group is adjusted to form a reconstructed surface in one or more additional phases.
Preferably, the method includes smoothing the reconstructed surface.
Preferably, the method includes applying an affine transformation to the reconstructed surface.
Preferably, the method includes a final stage in which each determined location is associated with a respective grid point, and the associated grid points are moved onto the determined locations while non-associated grid points are substantially not moved.
Preferably, the method includes estimating a measure of the volume responsive to the reconstructed surface.
Preferably, estimating the measure of the volume includes choosing an arbitrary point inside the grid and calculating the volumes of tetrahedrons defined by the arbitrary point and groups of three points on the grid which cover the entire grid surface.
There is further provided in accordance with a preferred embodiment of the present invention, apparatus for reconstructing a map of a volume from coordinates of a plurality of determined locations on a surface of the volume having a configuration, including a processor, which receives the coordinates and generates a grid of points defining a reconstruction surface in 3D space in proximity to the determined locations, and which defines a respective vector for each of the points on the grid, dependent on a displacement between one or more of the points on the grid and one or more of the locations, and which adjusts the reconstruction surface by moving each of the points on the grid responsive to the respective vector, so that the reconstruction surface is deformed to resemble the configuration of the surface of the volume.
Preferably, the apparatus includes a display screen for displaying the adjusted surface.
Preferably, the processor analyzes the adjusted surface to determine a characteristic of the volume.
Preferably, the apparatus includes a memory for storing the adjusted surface.
Preferably, the grid initially encompasses substantially all of the determined locations.
Preferably, the apparatus includes an imaging device for acquiring an image of the volume, and the processor defines the grid initially such that it resembles the image of the volume.
Preferably, the apparatus includes a memory library including a plurality of closed curves, and the processor defines the grid initially by choosing a closed curve from the memory library according to at least one characteristic of the volume.
Preferably, the processor generates and defines the reconstruction surface substantially independently of any assumption regarding a topology of the volume.
Preferably, the processor generates and defines the reconstruction surface substantially without reference to any point within the volume.
Preferably, the processor forms the adjusted grid in two stages: a rough adjustment stage and a flexible matching stage.
Preferably, in the rough adjustment stage, the processor moves each point on the grid toward a respective weighted center of mass of the determined locations, and locations closer to the point on the grid are given larger weight.
Preferably, the processor calculates the center of mass using a weight that is substantially proportional for each location to the inverse of the sum of a small constant and the distance between the point and the location raised to a power between 4 and 10.
Preferably, the constant is smaller than a precision of the location determination.
Preferably, in the flexible matching stage, the processor selects a respective grid point to associate with each of the determined locations.
Preferably, the selected grid point for each determined location includes a point on the grid that is closest to the location.
Preferably, in the flexible matching stage, the processor moves the selected grid points toward their respective, associated locations.
Preferably, the processor moves the selected grid points onto the associated locations.
Preferably, the processor moves non-selected grid points by an amount dependent on the movements of surrounding grid points.
Preferably, the amount of movement of the non-selected grid points is dependent on the movements of surrounding selected grid points.
Preferably, the amount of movement of each of non-selected grid points is calculated by the processor based on the distances from the surrounding selected grid points to the non-selected grid point.
Preferably, the amount of movement of the non-associated grid points is calculated by the processor based on an interpolation of the movements of surrounding selected grid points.
Preferably, the distances include geometrical distances. Preferably, the apparatus includes a probe, which is brought into engagement with the surface to determine the locations thereon.
Further preferably, the probe includes a position sensor which indicates the position of a tip of the probe.
Preferably, the sensor includes at least one coil.
Preferably, the sensor generates signals indicative of position and orientation of the sensor.
Alternatively or additionally, the probe includes a functional portion for acquiring a value of a physiological activity at the plurality of locations.
Preferably, the functional portion includes an electrode.
Preferably, the processor estimates a value of the physiological activity at the adjusted grid points.
Preferably, the processor estimates the value of the physiological activity based on the acquired values of the physiological activity at points surrounding the adjusted grid points.
Preferably, the processor estimates the value by interpolation from the acquired values responsive to deformation of the reconstruction surface.
Preferably, the apparatus includes a reference catheter for registering the determined locations relative to a frame of reference associated with the volume.
Preferably, the apparatus includes an ECG monitor for gating the operation of the probe so as to determine the points at a fixed phase of a cyclic movement of the volume.
There is further provided in accordance with a preferred embodiment of the present invention, a method of displaying values of a parameter which varies over a surface, including determining a value of the parameter at each of a plurality of points on the surface, and rendering an image of the surface to a display with a different degree of transparency in different areas of the surface, responsive in each of the areas to the value of the parameter at one or more points in the area.
Preferably, determining the value includes sampling a plurality of points and creating a map of the surface responsive thereto, and rendering the image includes rendering a graphic representation of the map.
Preferably, creating the map includes creating a three-dimensional map.
Preferably, determining the value includes determining a measure of reliability of the map in each of the areas.
Preferably, rendering the image includes rending one or more of the areas having a low measure of reliability relative to another one or more of the areas with a relatively greater degree of transparency.
Preferably, determining the measure of reliability includes determining a density of the sampled points.
Preferably, rendering the image includes defining a color scale and displaying a color associated with the value, at each of the plurality of points.
Preferably, the plurality of points includes sampled points and interpolated points, and determining the measure of reliability includes assigning a high reliability measure to the sampled points.
Preferably, determining the measure of reliability includes assigning measures of reliability to the interpolated points according to their respective distance from a closest sampled point.